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student that has at least 10 friends
parmenides51 2
N
by AylyGayypow009
Source: 2023 Greece JBMO TST P1
A class has 
students. Each group consisting of three of the students meet, and choose one of the other 
students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least 
friends.
students. Each group consisting of three of the students meet, and choose one of the other 
students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least 
friends.
2 replies
Interesting inequality
sealight2107 6
N
by TNKT
Source: Own
Let 
such that 
. Find the minimum value of:
such that 
. Find the minimum value of:
6 replies
truncated cone box packing problem
chomk 0
box : 48*48*32
truncated cone: upper circle(radius=2), lower circle(radius=8), height=12
how many truncated cones are packed in a box?
truncated cone: upper circle(radius=2), lower circle(radius=8), height=12
how many truncated cones are packed in a box?
0 replies
Dwarfes and river
RagvaloD 8
N
by AngryKnot
Source: All Russian Olympiad 2017,Day1,grade 9,P3
There are 
dwarfes with weight 
. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
dwarfes with weight 
. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
8 replies
Proving that these are concyclic.
Acrylic3491 1
N
by Funcshun840
In 
, points 
and 
are isogonal conjugates. The tangent to 
at and the tangent to 
at Q, meet at 
. 
intersects 
at 
. Prove that points ,, and are concyclic.
Any hints on this ?
, points 
and 
are isogonal conjugates. The tangent to 
at and the tangent to 
at Q, meet at 
. 
intersects 
at 
. Prove that points ,, and are concyclic.Any hints on this ?
1 reply
Concurrent Gergonnians in Pentagon
numbertheorist17 18
N
by Ilikeminecraft
Source: USA TSTST 2014, Problem 2
Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle gergonnians.
(a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent.
(b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.
(a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent.
(b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.
18 replies
Planes and cities
RagvaloD 11
N
by AngryKnot
Source: All Russian Olympiad 2017,Day1,grade 9,P1
In country some cities are connected by oneway flights( There are no more then one flight between two cities). City 
called "available" for city 
, if there is flight from to , maybe with some transfers. It is known, that for every 2 cities and exist city , such that and are available from . Prove, that exist city , such that every city is available for .
called "available" for city 
, if there is flight from to , maybe with some transfers. It is known, that for every 2 cities and exist city , such that and are available from . Prove, that exist city , such that every city is available for .
11 replies
Hard geometry
Lukariman 4
N
by Lukariman
Given circle (O) and chord AB with different diameters. The tangents of circle (O) at A and B intersect at point P. On the small arc AB, take point C so that triangle CAB is not isosceles. The lines CA and BP intersect at D, BC and AP intersect at E. Prove that the centers of the circles circumscribing triangles ACE, BCD and OPC are collinear.
4 replies
Three concurrent circles
jayme 0
Source: own?
Dear Mathlinkers,
1. ABC a triangle
2. 0 the circumcircle
3. Tb, Tc the tangents to 0 wrt. B, C
4. D the point of intersection of Tb and Tc
5. B', C' the symmetrics of B, C wrt AC, AB
6. 1b, 1c the circumcircles of the triangles BB'D, CC'D.
Prove : 1b, 1c and 0 are concurrents.
Sincerely
Jean-Louis
1. ABC a triangle
2. 0 the circumcircle
3. Tb, Tc the tangents to 0 wrt. B, C
4. D the point of intersection of Tb and Tc
5. B', C' the symmetrics of B, C wrt AC, AB
6. 1b, 1c the circumcircles of the triangles BB'D, CC'D.
Prove : 1b, 1c and 0 are concurrents.
Sincerely
Jean-Louis
0 replies
